A periodic orbit corresponds to a special type of solution for a dynamical system, namely one which repeats itself in time. A dynamical system
exhibiting a stable periodic orbit is often called an oscillator.

for all \(t\ .\) The period of this solution is defined to be the minimum such \(T\ .\) The image of the
periodicity interval \( [0,T] \) under \( x \) in the state space \( \mathbb{R}^n \) is called the
periodic orbit or cycle.

Limit Cycle

A periodic orbit \(\Gamma\) on a plane (or on a two-dimensional manifold)
is called a limit cycle if it is the \(\alpha\)-limit set or \(\omega\)-limit set of some point \(z\) not on the periodic orbit, that is, the set of accumulation points of either the forward or backward trajectory through \(z\ ,\) respectively, is exactly \(\Gamma\ .\) Asymptotically stable and unstable periodic orbits are examples of limit cycles.

Example (Guckenheimer and Holmes, 1983; Strogatz 1994)

The figure shows the periodic orbit which exists for the vector field
\[
\begin{matrix}
\frac{d x}{dt} = \alpha x-y-\alpha x(x^2+y^2) \\
\frac{d y}{dt} = x+\alpha y-\alpha y(x^2+y^2) ,
\end{matrix}
\]
where \(\alpha>0\) is a parameter. Transforming to radial
coordinates, we see that the periodic orbit lies on a circle with
unit radius for any \(\alpha>0\ :\)
\[
\frac{d r}{dt} = \alpha r(1-r^2), \qquad \frac{d \theta}{dt} = 1 .
\]
This periodic orbit is a stable limit cycle for \(\alpha>0\) and unstable limit cycle for \(\alpha<0\ .\) When \(\alpha=0\ ,\) the system above has infinite number of periodic orbits and no limit cycles.

Periodic Orbit for a Map

A periodic orbit with period \(k\) for a map
\[
x_{i+1} = g(x_i), \qquad x \in \mathbb{R}^n, \qquad n \ge 1
\]
is the set of \(k\) distinct points
\(\{p_j = g^j(p_0)| j=0,\cdots,k-1\}\) with
\(g^k(p_0) = p_0\) (Guckenheimer and Holmes, 1983). Here \(g^k\)
represents the composition of \(g\) with itself \(k\) times.
The smallest positive value of \(k\) for which this equality holds
is the period of the orbit. An example of a periodic orbit for a map is shown
in the figure.

Existence (or Non-Existence) of Periodic Orbits

It is sometimes possible to prove analytically that a periodic orbit exists
or cannot exist for a dynamical system using the following techniques.
Several of these apply for an autonomous planar vector field
\[
\frac{d x}{dt} = F(x,y), \qquad \frac{dy}{dt} = G(x,y), \qquad (x,y) \in \mathbb{R}^2.
\]

Index Theory

For an autonomous planar vector field, index theory can be used to
show that (Guckenheimer and Holmes, 1983):

Inside the region enclosed by a periodic orbit there must be at
least one equilibrium, i.e., a point where \(F(x,y)=G(x,y)=0\ .\)
If there is only one, it must be a
sink, source, or center. If all equilibria
inside the periodic orbit are hyperbolic, then there must be
an odd number \(2 m + 1 \ ,\) of which \(m\) are saddles and
\(m+1\) are sinks or sources.

This can be useful for showing that a periodic orbit does not
exist in a region of phase space: if the appropriate equilibria
are not present, a periodic orbit cannot exist.

Dulac's Criterion

For an autonomous planar vector field,
Dulac's criterion states (Guckenheimer and Holmes, 1983):

Let \(B(x,y)\) be a scalar function defined on a simply connected region
\(D \subset \mathbb{R}^2\) (so that \(D\) has no holes in it). If
\(\frac{\partial (B F)}{\partial x} + \frac{\partial (B G)}{\partial y}\)
is not identically zero and does not change sign in \(D\ ,\) then
there are no periodic orbits lying entirely in \(D\ .\)

Bendixson's Criterion

Dulac's criterion is a generalization of Bendixson's criterion, which corresponds
to \(B(x,y) = 1\) in the above result. These criteria can be useful for
showing that a periodic orbit does not exist in a region of phase space.

Poincare-Bendixson Theorem

If a trajectory enters and does not leave a closed and bounded region of
phase space which contains no equilibria, then the trajectory must
approach a periodic orbit as \(t \rightarrow \infty\ .\)

This can sometimes be used to establish the existence of a (stable) periodic
orbit for a planar vector field.

Liénard Systems

For nonlinear oscillators satisfying Liénard's equation
\[
\frac{d^2 x}{dt^2} + F \left(x,\frac{dx}{dt} \right) \frac{dx}{dt} + G(x) = 0, \qquad x \in \mathbb{R},
\]
the existence of a unique, stable limit cycle can be established under
appropriate general hypotheses on \(F\) and \(G\ .\)
For example, the damping coefficient \(F\) must be
negative near the phase space origin \(x = \frac{dx}{dt} = 0\) so
trajectories near the origin spiral outwards, and \(F\) must be
positive far away from the origin, so that trajectories far from the
origin spiral inwards. For a detailed discussion, see Jordan and Smith (1977).

Fast-Slow Planar Systems

For a fast-slow autonomous planar vector field
\[
\frac{dx}{dt} = F(x,y), \qquad \frac{dy}{dt} = \epsilon G(x,y), \qquad (x,y) \in \mathbb{R}^2, \qquad \epsilon << 1,
\]
simple geometrical nullcline analysis can suggest the
existence of a relaxation oscillation, a special type of
periodic orbit (Keener and Sneyd, 1998). The Poincare-Bendixson theorem can be used
to prove the existence of a periodic orbit in some cases, but this does not establish
that the orbit is a relaxation oscillation. Rigorous results for
relaxation oscillations are given in Grasman (1987) and Mishchenko et al. (1994); these make
use of geometric singular perturbation theory and go beyond the planar case. Fast-slow systems
can also have special periodic orbit solutions called canards, although these are not
robust to perturbations in planar systems.

Hilbert's 16th Problem

In 1900, David Hilbert famously posed 23 problems at the International
Congress of Mathematicians in Paris. His 16th problem involves
determining the number and location of limit cycles for an
autonomous planar vector field for which both \(F\)
and \(G\) are real polynomials of degree \(N\ .\) At present, this
problem has not been solved, but much progress has been made in the last 100+ years.
For example, it has been shown that the number of limit cycles for such a system is
finite. This and many other results are summarized in Ilyashenko (2002).

Gradient Flows

An autonomous vector field is called a gradient flow if it can be rewritten as
\[
\frac{d x}{dt} = -\nabla V(x), \qquad V:\mathbb{R}^n \rightarrow \mathbb{R},
\]
where the minus sign is included by convention, so that \(V(x)\) is a Lyapunov function for the system. Periodic orbits cannot exist for gradient flows (Guckenheimer and Holmes, 1983).

Sometimes a non-autonomous vector field with a small parameter
(including weakly nonlinear forced oscillations) can be rewritten
in a form which allows the method of averaging (over time)
to be applied to understand its dynamics. Most useful for the
present discussion is the result that the existence of a
hyperbolicequilibrium point of the resulting autonomous
equations implies the existence of a periodic orbit (possibly trivial, i.e., an equilibrium point) of the original non-autonomous system, with the same stability
properties as the equilibrium point (Guckenheimer and Holmes, 1983).

Finding Periodic Orbits for a Map

From our discussion above, each point \(p_j\) on a period-\(k\) periodic
orbit for a map is a fixed point for the map \(g^k\ .\)
Thus, one can find points on period-\(k\) periodic orbits by solving the
algebraic equation \(g^k(x) = x\) for \(x\ .\)
This may locate fixed points and points on periodic orbits with
periods less than \(k\ :\) for example, a fixed point with
\(g(x) = x\) is also a solution to \(g^k(x) = x\)
for any \(k\ .\) Even if the points on periodic orbits cannot be
found explicitly, analytical techniques might be used to prove that they must exist.

Periodic orbits can sometimes be found for a given vector field
using numerical methods. If a periodic orbit is stable, then forward
numerical integration of a trajectory with an initial condition in the periodic
orbit's basin of attraction will converge to the periodic orbit as
\(t \rightarrow \infty\ .\) Other methods can be used to numerically
find periodic orbits even if they are unstable. For example, the problem
of finding (stable or unstable) periodic orbits for an
autonomous vector field can be reformulated so that a
variant of the Newton-Raphson algorithm can be
applied; one numerically solves \(\phi_T(x) - x = 0\)
for \(x\) and \(T\ ,\) where \(\phi_T(x)\) is the location
of a trajectory starting at the point \(x\) after a time \(T\)
(Parker and Chua, 1989). More robust numerical methods are based on a boundary value problem on the unit interval for the periodic solution \(x(t)=u(t/T) \ :\)

\[
u'-Tf(u)=0, u(0)=u(1), \Psi[u]=0,
\]

where \( \Psi \) is a phase condition selecting one periodic solution among infinitely
many periodic solutions corresponding to the same periodic orbit but having different initial points
(Doedel, Keller, and Kernevez, 1991).
This BVP should then be approximated by a proper finite-dimensional discretization (e.g.,
via orthogonal collocation with piecewise-polynomial functions) and solved for
(the discretization of) \( u \) and \( T \ .\)

The Newton-Raphson algorithm (or other root finding methods) can be directly applied
to find points on periodic orbits for a map: one just needs to find roots of the
equation \(g^k(x) - x = 0\) for the period \(k\) of
interest.

Stability of a Periodic Orbit

The stability of a periodic orbit for an autonomous vector field
can be calculated by considering the Poincare map which replaces the flow
of the \(n\)-dimensional continuous vector field with an
\((n-1)\)-dimensional map (Guckenheimer and Holmes, 1983).
Specifically, an \((n-1)\)-dimensional
surface of section \(\Sigma\) is chosen such that the flow is
always transverse to \(\Sigma\) (see figure).
Let the successive intersections in a given direction of the solution
\(x(t)\) with \(\Sigma\) be denoted by \(x_i\ .\) The
Poincare map
\[
x_{i+1} = g(x_i)
\]
determines the \((i+1)\)-th intersection of the trajectory with \(\Sigma\)
from the \(i\)-th intersection. A periodic orbit of an
autonomous vector field corresponds to a fixed point \(x_f\) of this
Poincare map, characterized by \(g(x_f) = x_f\ .\) The linearization of
the Poincare map about \(x_f\) is
\[
\xi_{i+1} = {\rm D}g(x_f) \xi_i.
\]
If all eigenvalues of \({\rm D} g\) have modulus less than unity, then
\(x_f\) (and thus the corresponding periodic orbit) is
asymptotically stable. If any eigenvalues of \({\rm D} g\) have
modulus greater than unity, then \(x_f\) (and thus the
corresponding periodic orbit) is unstable. The stability
properties of a periodic orbit are independent of the cross section
\(\Sigma\) (Wiggins 2003). If \( x_f \) is stable
then it is an attractor of the Poincare map, and the
corresponding periodic orbit is an attractor of
the vector field.

Example (continued) (Guckenheimer and Holmes 1983, Strogatz 1994)

For the Example above, the radial line
given by \(\theta=0\) is a Poincare section, parameterized by
\(r\ .\) The corresponding Poincare map \(r_{i+1} = g(r_i)\) along
this section may be found by explicitly integrating the vector field:
\[
g(r_i)=\left[ 1+ e^{ - 4 \pi \alpha}(r_i^{-2} -1 ) \right]^{-1/2},
\]
with fixed point \(r_f=1\) corresponding the periodic orbit.
Linearizing, we find \(g'(r_f)=e^{ - 4 \pi \alpha}\ .\)
So, the periodic orbit is stable for any
\(\alpha>0\) and is unstable for any \(\alpha<0\ .\)

An alternative way to determine the stability of a periodic
orbit is to use Floquet theory, which involves the
time-dependent (and \(T\)-periodic) vector field linearized around
the periodic orbit. Solutions to these linearized equations are
used to define \(n\) Floquet multipliers characterizing the
growth or decay of perturbations to the periodic orbit. It can be shown that the
\((n-1)\) eigenvalues of \({\rm D}{g}\) are equal to
\((n-1)\) of the Floquet multipliers of the periodic orbit; the
remaining Floquet multiplier is equal to unity and
corresponds to a perturbation along the periodic
orbit (Guckenheimer and Holmes, 1983). The determination of Floquet
multipliers or the eigenvalues of \({\rm D}{g}\) typically must
be done numerically.

Given a point \(x_f\) on the periodic orbit \(\Gamma\) as discussed above, the eigenvalues of the matrix \({\rm D}g(x_f)\) can be used to partition the \((n-1)\)-dimensional subspace \(\Sigma\) into a direct sum of subspaces \(\Sigma^s \oplus \Sigma^c \oplus \Sigma^u\ ,\) corresponding to eigenvalues with modulus less than 1, equal to 1, and greater than 1, respectively. If sections \(\Sigma_x\) are chosen to vary continuously over different base points \(x \in \Gamma\ ,\) then concatenations of the corresponding subspaces \(\Sigma_x^s, \Sigma_x^c, \Sigma_x^u\) form vector bundles over \(\Gamma\ .\) Stable, center, and unstable manifolds of \(\Gamma\) can be defined as graphs over these vector bundles.

For a non-autonomous vector field \(\frac{dx}{dt} = f(x,t)\)
with \(f(x,t) = f (x,t+\tau)\) for some \(0<\tau<\infty\ ,\)
the calculation of the stability properties of a periodic orbit with
period \(T = \frac{p \tau}{q}\ ,\) where \(p\) and \(q\) are integers (see Arnold tongues),
can be done by considering a stroboscopic map which takes
\[
x(t) \rightarrow x\left(t + \frac{p \tau}{q} \right).
\]
The stability
properties follow from the eigenvalues of this map, as above.

To determine the stability properties of a periodic orbit for a
mapping \(x_{i+1} = g(x_i)\ ,\) one can exploit the fact that a point \(p_0\) on a
period-\(k\) periodic orbit of the map \(g\) is a fixed point of
the map \(g^k\ .\) The stability properties of this fixed point
of \(g^k\) are the same as the stability properties of the periodic orbit
of the map \(g\) (Guckenheimer and Holmes, 1983).

A bifurcation is a qualitative change in the behavior of a
dynamical system as a system parameter is varied. This
could involve a change in the stability
properties of a periodic orbit, and/or the creation or destruction
of one or more periodic orbits. Bifurcation analysis can thus provide
another (analytical or numerical) method for establishing the
existence or non-existence of a periodic orbit.

Blue-Sky Catastrophe, in which a periodic orbit of large period appears "out of a blue sky" (actually, the orbit is homoclinic to a saddle-node periodic orbit).

These bifurcations result in the appearance or disappearance of periodic orbits, depending on the direction in which the bifurcation parameter is varied. The (dis)appearing orbits may be stable or unstable, depending, among other factors, on whether the bifurcations are subcritical or supercritical.

Periodic Orbits and Chaos

As a system parameter is varied, chaos can appear via an
infinite sequence of period doubling bifurcations of
periodic orbits. This is known as the Feigenbaum phenomenon or
the period doubling route to chaos (Ott, 1993).
Moreover, a chaotic attractor typically has a dense set of
unstable periodic orbits embedded within it. Suitable averages
over such periodic orbits can be used to approximate descriptive
quantities for chaotic attractors such as Lyapunov exponents
and fractal dimensions (Chaos Focus Issue, 1992). Such
periodic orbits can sometimes be stabilized (and the chaos thus
suppressed) through small manipulations of a system parameter, an
approach called controlling chaos (Ott 1993).